Abelian groups and Homology groups

Given an abelian group , and a natural number , does there exist a space X such that ?
Where is the n-dimensional homology group of X.
I know that for every group , there is a 2-dimensional cell complex such that the fundamental group of is isomorphic to .
Can we generalize this result to the homology groups, when is abelian? If yes, how do you prove this

The 1st homology group is the abelianization of the fundamental group, so as you said, you can find a space with arbitrary H1 (although I believe the group G needs to be finitely presented). The suspension SX of a space X has the property that for all k, which will finish it up for you. This is exercise 2.2.32 in Hatcher, and is proved with the Mayer-Vietoris sequence.